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Works  on  Drawing 

by  Frank  E.  Mathewson 

Published  by  the  Taylor-Holden  Company 

Springfield,  Mass. 

Notes  for  Mechanical  Drawing  -     -     -     $1.25 

Brief  Course  in  Machine  Drawing     -     -     -     -  .60 
Architectural  Drawing  Plates 

Folio  1.    Details  of  Construction  (10  plates)  .75 

Folio  2.     Brickwork  and  Masonry  (10  plates)  .75 

Mechanical  Drawing  Alphabets,  (set  of  six)  .10 

Perspective  Sketching  from  Working  Drawings  1.00 

Others  in  preparation. 


PERSPECTIVE  SKETCHING 

From  Working  Drawings 

BY 

FRANK    E.   MATHEWSON 

ASSISTANT  PRINCIPAL  :  THE  TECHNICAL 
HIGH   SCHOOL,  CLEVELAND,  OHIO 


THE  TAYLOR-HOLDEN  COMPANY 

SPRINGFIELD,  MASS.       -        1910 


-Second  Edition — 
—October  1910— 


COYPRIGHT,  1908 

BY 

FRANK  E.  MATHEWSON. 


The  F.  A.  Bassette  Co.,  Printers,  Springfield,  Mass 


PREFACE 

The  course  in  perspective  sketching  from  working 
drawings,  outlined  in  this  manual,  is  a  result  of  the 
author's  experience  in  teaching  this  branch  of  drawing 
to  first  and  second  year  pupils  in  the  Technical  High 
School,  Springfield,  Massachusetts.  It  was  found 
that  many  pupils  who  could  readily  make  working 
drawings  from  models  or  sketches,  did  not  easily 
recognize  similar  forms  when  embodied  in  more  or  less 
complicated  working  drawings.  A  series  of  exercises  and 
problems  in  perspective  sketching  from  working  draw- 
ings without  the  use  of  models,  was  introduced  into 
the  course  in  drawing.  The  result  was  so  satisfactory 
that  gradually  mere  time  was  devoted  to  this  subject, 
and  it  has  been  found  that  students  progress  more 
rapidly  in  the  reading  and  making  of  complicated 
working  drawings  after  completing  this  part  of  the 
course.  The  great  advantage  to  the  student  in  having 
the  power  to  translate  working  drawings  into  perspec- 
tive sketches  is  that  it  gives  him  the  ability  to  make  clear 
in  this  way  the  meaning  of  such  drawings  to  those  un- 
familiar with  them.  It  is  not  within  the  province  of 
this  manual  to  enter  into  a  discussion  of  the  many 
theories  and  various  principles  of  perspective  drawing. 
Aside  from  a  plain  statement  of  facts  connected  with 
the  fundamental  principles  of  perspective,  the  manual 
contains  exercises  and  problems  carefully  selected  in 
order  to  make  suitable  application  of  these  principles 
in  perspective  sketching  from  working  drawings. 

August,  1908. 


Contents 


TEXT 

SECTION  PAGE 

1  Introduction 7 

2  Definitions 8 

3  Measurements 12 

4  The  Square  in  Perspective 13 

5  The  Cube  in  Perspective 15 

6  Angle  Plate 19 

7  The  Circle  in  Perspective 21 

8  The  Cylinder  in  Perspective 27 

9  Estimating  Measurements  in  Perspec- 

tive   28 

10  The  Elliptical  Protractor 30 

11  The  Rectangular  Prism  in  Perspec- 

tive    34 

12  The  Equilateral  Triangle  in  Perspec- 

tive   39 

13  The  Hexagon  in  Perspective 41 

14  The  Square  Pyramid  in  Perspective .  .  45 

15  The  Cone  in  Perspective 47 

16  Perspective    Sketching   from   Work- 

ing Drawings 48 

FULL  PAGE  PLATES 

NUMBER  PAGE 

I     Illustrating  "  Definitions" 9 

II     Parallel  Planes 11 

III     The  Square  in  Perspective 14 

.  IV     Cubes  in  Perspective 16 

V     Angle  Plate;  Drawing  and  Sketch  . .  18 


VI  Angle  Plate  in  six  positions 20 

VII  The  Circle  in  Perspective 22 

VIII  Parallel  Circular  Planes 24 

IX  Cylinders  in  Perspective 26 

X  Rectangular  Prisms  in  Perspective. .  35 

XI  Rectangular    Prisms    and    Card    in 

Perspective 37 

XII  The  Equilateral  Triangle  in  Perspec- 
tive   40 

XIII  The  Hexagon  in  Perspective 42 

XIV  The  Square  Pyramid  in  Perspective .  .  44 
XV  The  Cone  in  Perspective 46 

XVI  Hollow  Sleeve 51 

XVII  Bush  for  Bearings 53 

XVIII  Grooved  Block 55 

XIX  V  Block 57 

XX  Crank 59 

XXI  Offset  Standard 61 

XXII  Angle  Brace 63 

XXIII  Intersecting  Square  Prisms 65 

XXIV  Crank 69 

XXV  Pipe  Tee 71 

XXVI  Shaft-Hanger 73 

XXVII  Tool  Post.     Working  Drawing 74 

XXVIII  Tool  Post  in  Perspective 76 


Perspective  Sketching  from 
Working  Drawings 


1.  Introduction.  To  the  engineer,  architect,  designer 
and  expert  craftsman,  drawing  is  essential  as  a 
means  of  graphic  expression  of  thought.  Through  its 
use,  they  are  able  to  design,  clarify  and  record  their 
inventive  suggestions.  Without  it,  any  advancement 
in  their  professional  work  is  practically  impossible. 
A  course  in  drawing  should  involve  such  a  study  of  the 
various  principles  and  methods  of  representation,  and 
their  application  to  the  arts,  that  it  shall  equip  the 
student  with  all  the  necessary  information  from  the 
drawing  side,  so  that  he  will  be  able  to  draw  under- 
standingly,  to  grasp  a  difficult  problem,  to  get  it  on 
paper  quickly,  and  make  a  drawing  that  shall  be 
acceptable  in  any  drafting  room  or  shop. 

"Mechanical  or  constructive  drawing  is  the  language 
of  the  workshop,"  is  a  familiar  statement.  Nasmyth 
says,  "  Mechanical  drawing  is  the  alphabet  of  the  engi- 
neer. Without  this  he  is  merely  a  'hand;'  with  it,  he 
indicates  the  possession  of  a  head." 

Equally  important  with,  and  a  very  necessary 
adjunct  to  the  knowledge  of  mechanical  drawing  is 
the  ability  to  readily  make  freehand  sketches,  either 
in  the  form  of  working  drawings  or  in  perspective. 
It  is  often  necessary  to  explain  working  drawings  to 
those  unfamiliar  with  them.  To  be  able  to  quickly 
translate  the  working  drawing  into  a  perspective 
sketch  of  the  object,  without  the  use  of  the  model,  in 
order  to  make  clear  the  thing  represented,  is  an  accom- 
plishment with  which  every  draughtsman  should  be 
familiar.  The  ability  to  draw  and  sketch  readily  is 


PERSPECTIVE  SKETCHING 


thus  becoming  more  and  more  a  necessary  accomplish- 
ment of  considerable  importance  to  those  engaged  in 
the  various  crafts,  and  while  it  is  not  likely  that  every 
one  will  become  an  artist,  it  is  possible  that  all  may  be 
taught  to  draw  well  through  a  thorough  understanding 
of  principles.  Once  learned  the  principles  will  fall  into 
their  proper  place.  The  draughtsman  will  do  his  best 
work  only  when  he  can  record  his  impressions  intui- 
tively and  correctly,  the  rules  and  principles  of  per- 
spective being  applied  almost  unconsciously. 

2.  Definitions.  PLATE  I.  PERSPECTIVE  is  the  sci- 
ence of  representing  in  picture  form  the  position  of 
lines  which  represent  objects  as  they  appear  under 
certain  more  or  less  definite  conditions. 

A  PICTURE  may  be  described  as  a  drawing  which 
by  its  lines  represents  an  object  as  it  would  appear  if 
viewed  through  a  window,  thus  giving  a  clear  im- 
pression of  its  form,  size  and  position  in  space. 

There  are  several  familiar  phenomena  of  perspec- 
tive which  must  be  particularly  observed  in  formulating 
principles  for  perspective  sketching. 

Receding  parallel  lines  seem  to  converge  toward  a 
common  vanishing-point.  A  familiar  illustration  of  this 
may  be  found  in  looking  down  a  straight  railway  track 
where  the  rails,  telegraph  wires,  fences,  etc.,  all  seem  to 
converge  to  a  point  which  is  directly  in  front  of  and  on 
a  level  with  the  eye.  Many  similar  illustrations  may 
easily  be  obtained  by  looking  at  any  series  of  re- 
treating parallel  lines. 

The  farther  an  object  is  from  the  observer,  the  smaller 
it  appears  to  be.  This  will  be  noticed  in  the  same  illus- 
tration of  the  railway  tracks,  for  in  looking  at  the  spaces 
between  the  telegraph  poles  or  the  railway  ties,  the 
nearer  spaces  appear  the  larger,  although  they  are  the 
same  distance  apart  as  those  which  are  farther  from  the 
spectator. 


FROM  WORKING  DRAWINGS 


9 


PLATE  I. 


1O  PERSPECTIVE  SKETCHING 

The  picture  of  that  point  toward  which  the  parallel 
lines  retreating  in  that  direction  seem  to  meet  is  called 
the  VANISHING-POINT.  (V.  P.) 

The  STATION-POINT  (S.  P.),  sometimes  called  the 
"point  of  sight,"  locates  the  position  of  the  observer's 
eye. 

The  PICTURE-PLANE  (P.  P.)  is  an  imaginary,  trans- 
parent plane,  placed  vertically  in  front  of  the  observer, 
through  which  he  is  supposed  to  be  looking  and  on  the 
surface  of  which  the  picture  is  represented. 

The  CENTER  OF  VISION  (C.  V.)  is  that  point  on 
the  picture -plane  directly  in  front  of  the  place  where 
the  observer  is  looking,  and  is  at  the  foot  of  a  perpen- 
dicular from  the  station-point  to  the  picture-plane.  It 
should  not  be  thought  of  as  the  center  of  the  picture, 
as  if  the  two  were  the  same,  for  they  are  not.  But  a  small 
part  of  that  which  is  seen  may  be  included  in  the  limits 
of  the  sketch,  and  that  part  may  lie  principally  either 
above  or  below  the  horizon. 

The  LINE  OF  SIGHT  is  the  line  from  5.  P.  to  C.  V. 
It  is  perpendicular  to  the  horizon  line  and  to  the  picture- 
plane  at  C.  V. 

The  HORIZON  in  a  drawing  is  the  picture  of  that 
line  which  is  infinitely  distant  from  the  observer,  and 
opposite  the  level  of  his  eye.  It  may  be  considered  as 
the  farther  edge  of  a  horizontal  plane  which  is  perpen- 
dicular to  the  picture-plane.  It  is  sometimes  called 
the  eye-level  line.  As  all  parallel  lines  have  a  common 
vanishing-point  on  the  horizon  or  eye-level  line,  so 
must  all  parallel  planes  have  a  common  vanishing- 
line.  This  is  illustrated  in  PLATE  II.  If  we  represent 
a  very  large  square  prism  extending  above  and  below 
the  horizon,  or  eye-level  line,  and  then  trace  the  outlines 
of  various  planes,  beginning  at  the  bottom  of  the  prism, 
at  intervals,  until  the  top  of  the  prism  is  reached,  it  may 
be  plainly  seen  that  all  of  these  planes  vanish  in  the 


FROM  WORKING  DRAWINGS 


II 


12  PERSPECTIVE  SKETCHING 

horizon  or  eye-level  line.  It  will  also  be  noted  that  the 
distance  between  the  front  and  back  corners  of  the 
planes  decreases  as  they  approach  the  eye-level  line. 

3.  Measurements.  All  perspective  measurements 
must  be  purely  relative  because  objects  change  their 
appearance  according  to  their  distance  from  the  obser- 
ver. It  is  impossible  to  tell,  until  proper  tests  are 
applied  to  the  drawing,  whether  an  object  in  a  picture 
is  represented  the  right  size  or  not.  Making  a  "  drawing 
to  scale"  has  not  the  same  meaning  in  perspective  that 
it  has  in  mechanical  drawing,  and  in  sketching,  all 
measurements  can  be  better  estimated  than  scaled. 
After  a  little  practice,  it  is  possible  to  make  such 
estimates  with  a  very  reasonable  degree  of  accuracy. 
When  it  is  necessary  to  make  a  perspective  drawing 
from  the  working  drawing,  or  to  represent  a  form  which 
is  not  at  hand  to  sketch  from,  the  draughtsman  must 
depend  wholly  upon  his  knowledge  of  the  appearance 
of  forms  which  has  been  gained  by  experience  in  sketch- 
ing, together  with  an  understanding  of  perspective 
principles.  A  rough  sketch  is  outlined,  and  to  this, 
perspective  tests  are  applied  in  order  to  correct  the 
inaccuracies  of  the  work  and  to  make  the  drawing  look 
as  if  it  were  made  on  the  transparent  picture-plane 
between  the  draughtsman  and  the  object.  The  exact 
positions  of  the  vanishing-points  need  not  be  considered, 
although  it  is  necessary  to  know  where  they  ought  to 
be  if  the  estimate  needs  correction.  This  is  also  true 
in  regard  to  the  exact  measurements  of  the  different 
parts.  The  true  impression  which  the  drawing  will 
make  upon  the  observer  is  the  thing  desired  and  toward 
which  the  draughtsman  must  ever  strive. 

One  of  the  most  frequently  used  methods  of  measur- 
ing in  perspective  is  by  means  of  the  intersection  of 
two  or  more  series  of  parallel  lines,  and  this  method 
depends  upon  the  geometric  principle  that  spaces 
between  parallels  are  equal.  Figure  1  shows  an  appli- 


N 


FROM  WORKING  DRAWINGS 


13 


FIGURE   1. 

cation  of  the  measurement  by  parallels,  for  if  the  three 
edges,  ab,  ac,  and  ad,  are  determined,  then  the  other 
edges  parallel  with  these,  which  complete  the  picture  of 
the  cube,  may  be  represented  by  drawing  lines  appar- 
ently parallel  and  converging  with  the  first  three  lines. 
These  lines  also  determine  the  vertical  height  of  the 
edges,  ch,  dg,  and  ef.  Excellent  examples  of  the 
method  of  measuring  by  parallels  are  illustrated  in 
PLATES  V,  XVIII  and  XIX. 

4.  The  Square  in  Perspective.  PLATE  III.  With 
little  or  no  knowledge  of  perspective  principles  one  may 
make  a  fairly  good  sketch  of  a  square,  for  almost  any 
four  lines  that  enclose  a  polygon,  will  be  the  possible 
picture  of  a  square  seen  from  some  point  of  view.  The 
perspective  conditions  may  not  always  be  good  ones  to 
use  for  a  picture  of  the  square,  but  they  will,  however, 
be  true.  In  PLATE  III,  the  horizontal  line  from  a, 
Figure  1,  to  a,  Figure  3,  represents  the  lower  edge  of 
the  picture-plane  which  is  assumed  to  be  perpendicular 
to  the  ground-plane  that  vanishes  in  the  horizon  or 
eye-level  line.  In  each  figure,  the  line,  ab,  which  is  the 


14 


PERSPECTIVE  SKETCHING 


PLATE  III. 


FROM  WORKING  DRAWINGS  15 

front  edge  of  the  square,  abed,  is  parallel  with  the  picture- 
plane.  The  diagonal,  hf,  of  the  inner  square,  efgh,  is  also 
parallel  with  the  picture-plane,  thus  making  the  edges 
of  the  square,  efgh,  at  45  degrees  to  the  picture-plane. 
In  Figure  1,  both  squares  are  represented  parallel  with 
the  ground-plane  and  directly  in  front  of  the  observer. 
In  Figure  2,  they  are  moved  to  the  right  and  inclined 
at  an  angle  of  about  45  degrees  to  the  ground-plane. 
In  Figure  3,  the  squares  are  shown  in  a  vertical  position, 
perpendicular  to  the  ground-plane  and  moved  farther 
away  from  the  spectator.  In  the  application  of  the 
principles  illustrated  in  this  plate,  it  is  not  necessary 
that  the  horizon  line  or  vanishing-point  be  located  on 
the  drawing.  The  direction  of  the  lines  of  the  square 
can  always  be  determined  by  their  relation  to  each 
other. 

PROBLEM  1.  Make  sketches  of  a  square  in  per- 
spective, using  PLATE  III  as  a  guide  in  arranging  the 
drawing,  but  showing  the  square  in  four  positions, 
none  of  which  shall  be  exactly  like  those  in  PLATE  III. 

5.  The  Cube  in  Perspective.  PLATE  IV.  The  cube 
is  the  most  simple  in  form  of  the  geometric  solids,  and  as 
nearly  all  forms  may  be  analyzed  by  using  the  cube  as 
a  unit,  it  is  necessary  that  the  student  should  be  able 
to  determine  the  best  perspective  position  in  which 
to  represent  it. 

The  following  rules,  based  on  perspective  principles 
already  stated,  are  to  be  used  in  sketching  the  cube  in 
perspective. 

Retreating  parallel  lines  in  ike  object  all  converge  to 
a  common  vanishing-point  in  the  drawing. 

Vertical  lines  in  the  object  appear  as  vertical  lines 
in  the  drawing. 


16 


PERSPECTIVE  SKETCHING 


—  C(/oe$  in  Perspective  — 


PLATE  IV. 


FROM  WORKING  DRAWINGS  17 

The  nearest  line  of  a  cube  in  perspective  is  the  longest 
line  in  the  drawing,  and  all  retreating  parallel  lines  con- 
verge from  this  line  to  their  common  vanishing-point  in 
the  horizon  line. 

The  vanishing-points  should  be  considered  as 
being  at  a  great  distance  from  the  nearest  line  of  the 
cube  in  order  to  show  but  slight  convergence  in  each 
series  of  parallel  retreating  lines. 

A  good  position  to  assume  in  making  a  sketch  from 
a  working  drawing,  when  using  the  cube  as  a  unit,  is 
that  where  the  vertical  faces  are  equally  inclined  from 
the  picture-plane  and  the  top  face  far  enough  below 
the  eye  so  that  the  drawing  will  show  the  apparent 
length  of  the  short  diagonal  equal  to  about  one- 
third  of  the  height  of  the  front  vertical  edge.  The 
length  of  the  long  diagonal  should  be  little  less  than  one 
and  one-half  times  the  front  vertical  edge. 

Figure  1 ,  in  PLATE  IV,  represents  a  cube  in  such  a 
position.  After  determining  the  vertical  height  of  the 
cube,  ab,  and  drawing  the  top  edges,  ac  and  a/,  of  in- 
definite length,  the  back  corner,  g,  may  be  located  about 
one-third  of  the  height,  ac,  vertically  above  a.  Then 
drawing  the  long  diagonal,  fc,  through  m  which  is  just 
a  little  nearer  g,  than  it  is  to  a,  the  corners,  /  and  c,  are 
definitely  fixed.  After  drawing  eg  and  fg,  it  will  be  noted 
that  they  have  a  slight  convergence  toward  the  lines,  af 
and  ac.  The  lines,  bd,  be,  dh,  and  eh,  are  then  drawn, 
and  care  must  be  taken  to  have  them  apparently  con- 
verge with  the  lines  of  the  top  face,  toward  the  same 
vanishing-point  which  is  somewhere  in  the  eye-level 
line  at  an  indefinite  distance  from  the  front  vertical 
edge  of  the  cube.  Points,  e  and  d,  are  directly  under  / 
and  c,  and  parallel  vertical  lines  connecting  fe  and  ad 
complete  the  drawing.  Figure  2,  in  this  plate  shows 
the  construction  of  square  prisms  based  on  the  cube 


18 


PERSPECTIVE  SKETCHING 


r-  — 


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£.  A 


^«a 

1  1 

•f 

I 

PLATE  V. 


FROM  WORKING  DRAWINGS  19 

as  a  unit.  The  four  cubes  in  line  illustrate  the  con- 
struction of  a  square  prism  in  a  horizontal  position, 
and  the  two  cubes,  one  above  the  other,  show  the  con- 
struction of  a  vertical  square  prism. 

PROBLEM  2.  Make  a  sketch  similar  to  Figure  1, 
PLATE  IV,  of  a  cube,  the  nearest  edge  of  which  appears 
to  be  3"  long. 

6.  Angle  Plate.  PLATE  V.  Figures  1  and  2  are 
working  drawings  representing  the  two  elevations  of 
an  angle  plate.  By  extending  the  lines,  fh  and  ak, 
to  b  in  Figure  1,  the  outlines  of  both  Figures  1  and  2 
represent  two  views  of  a  cube  which  may  be  used  as  a 
unit  in  making  the  perspective  sketch  of  the  angle 
plate.  Sketch  the  perspective  of  the  cube,  Figure  3, 
with  the  front  vertical  edge,  ab,  equal  to  the  height  of 
the  angle  plate,  3£",  and  the  retreating  parallel  edges 
located  as  represented  in  Figure  1 ,  PLATE  IV.  The  lines 
of  the  angle  plate  which  are  parallel  with  the  outside 
lines  should  then  be  sketched,  starting  from  the  point 
k,  which  is  f  "  above  a,  and  converging  toward  the  same 
indefinite  vanishing-points  of  the  lines  with  which  they 
are  parallel  in  the  working  drawing.  The  point,  m,  is 
at  the  intersection  of  lines  drawn  from  k  and  h,  and  n 
is  at  the  intersection  of  lines  drawn  from  i  and  i.  The 
line  mn,  is  at  the  intersection  of  the  two  inner  surfaces 
of  the  angle  plate.  Estimate  the  position  of  i  and  2, 
sketch  the  vertical  lines,  i-j  and  3-4,  and  the  retreating 
converging  lines,  1-5  and  2-6.  These  lines  locate  the 
positions  in  the  lines,  hi  and  kl,  of  the  ends  of  the  slant- 
ing lines  forming  the  brace.  The  vertical  lines  of  the 
brace  are  then  sketched,  locating  points  7  and  8,  from 
which  the  lower  lines  of  the  brace  are  continued  to  the 
line  kl. 

The  six  figures  in  PLATE  VI,  represent  the  angle 
plate  -  in  as  many  different  positions.  The  method 
illustrated  in  PLATE  V,  that  of  using  a  perspective 


2O 


PERSPECTIVE  SKETCHING 


-Angle  P/afe  — 


PLATE  VI. 


FROM  WORKING  DRAWINGS 


21 


FIGURE  2. 

cube  for  the  unit  of  construction,  is  followed.  In  each 
figure,  the  construction  by  the  intersections  of  the  series 
of  parallel  lines  converging  in  either  direction  from  the 
nearest  line  toward  common  but  indefinite  vanishing- 
points,  and  the  location  of  the  cube,  either  above  or 
below  the  eye-level,  is  clearly  illustrated. 

PROBLEM  3.  Make  perspective  sketches  showing 
two  different  positions  of  the  angle  plate,  of  which 
Figure  2,  is  the  working  drawing. 


7.  The  Circle  in  Perspective.  PLATE  VII.  The  cir- 
cle in  perspective  may  have  three  possible  forms,  de- 
pending altogether  on  its  position  in  regard  to  the 
observer.  If  the  plane  of  the  circle  is  parallel  with  the 
picture-plane,  then  the  result  will  be  a  circle.  If  the 
plane  of  the  circle  is  perpendicular  to  the  picture-plane 
and  directly  in  front  of  the  observer,  so  that  he  is  look- 
ing tangent  to  the  plane  of  the  circle,  then  the  result  is 
a  straight  line.  If  moved  in  the  slightest  degree  above  or 
below  the  eye-level  or  to  the  right  or  left  of  the  center 
of  vision,  the  picture  will  be  an  ellipse,  which  is  the  most 
common  form  of  the  perspective  drawing  of  the  circle. 


22 


PERSPECTIVE  SKETCHING 


PLATE  VII. 


FROM  WORKING  DRAWINGS  23 

Figure  1,  shows  a  circle  lying  in  a  horizontal 
position  and  enclosed  in  two  similar  squares,  one  with 
edges  parallel  with  the  picture-plane,  and  the  other  with 
edges  at  45  degrees  to  the  picture-plane.  The  pictures 
of  the  squares  are  different,  while  the  circle  remains 
unchanged  in  appearance  in  regard  to  the  position  of 
the  squares,  and  it  would  not  change  its  appearance  if 
the  squares  were  represented  at  other  angles.  It  may 
be  noted  in  Figure  2,  that  the  axes  of  the  ellipses  in 
perspective  do  not  coincide  with  the  diameters  or 
diagonals  of  the  squares  enclosing  them.  If  in  Figure  1 , 
the  line,  hg,  is  assumed  to  be  an  axis  of  revolution  per- 
pendicular to  the  plane  of  the  circle  and  squares,  the 
major  axis  of  the  ellipse  is  perpendicular,  and  the  minor 
axis  of  the  ellipse  coincides  with  this  axis  of  revolution 
of  the  circular  plane.  This  illustrates  the  following 
rule  which  is  of  general  application  in  sketching  the 
circle. 

"An  ellipse  which  is  the  picture  of  a  circle  should 
have  its  minor  axis  coincident  with  a  line  which  is  the 
picture  of  the  axis  of  revolution  of  the  circular  plane.  The 
major  axis  of  the  ellipse  should  be  perpendicular  to  the 
line  representing  the  axis  of  revolution  of  the  circular 
plane." 

In  Figure  2,  the  axes  of  revolution  of  the  two  ver- 
tical circular  planes  are  the  lines,  ab  and  gh.  The  minor 
axes,  ef  and  kl,  coincide  with  the  axes  of  revolution  and 
the  major  axes,  cd  and  ij,  are  perpendicular  to  them. 

There  is  one  important  factor  which  must  be 
noted  in  drawing  the  picture  of  concentric  circles.  In 
Figure  3,  each  circle  is  circumscribed  by  a  square,  and 
both  squares  have  the  same  picture  of  the  center,  but 
the  major  axes  of  the  ellipses  representing  the  circles 
do  not  coincide  with  the  diameter  of  the  square  or  with 


24 


PERSPECTIVE  SKETCHING 


C/rcv/ar  Pfones 


PLATE  VIII. 


FROM  WORKING  DRAWINGS 


FIGURE  3. 


each  other.  This  is  because  of  the  fact  that  objects 
appear  smaller  the  farther  they  are  away.  The  dis- 
tances between  the  curves  are  equal  at  the  ends  of  the 
major  axes,  because  these  distances  are  parallel  with  the 
picture-plane,  but  the  space  at  the  extreme  farther  end 
of  the  minor  axis  appears  less  than  that  at  the  nearer 
end.  This  difference  may  not  always  be  apparent  and 
it  depends  upon  the  relative  sizes  of  the  circle  and  the 
position  of  their  planes.  If  a  subject  has  many  con- 
centric circles  and  the  difference  in  size  is  not  very  great, 
it  is  allowable  to  ignore  this  difference,  which  simplifies 
the  construction  of  the  sketch. 

PLATE  VIII,  illustrates  the  different  apparent 
widths  of  parallel  circular  planes  as  they  vary  in  dis- 
tance from  the  eye-level  or  horizon  line.  Note  the  fact 
that  the  farther  away  from  the  horizon  line,  the  greater 
is  the,  length  of  the  minor  axes  in  proportion  to  the 
length  of  the  major  axes  of  the  ellipses. 


26 


PERSPECTIVE  SKETCHING 


PLATE  IX. 


FROM  WORKING  DRAWINGS 


27 


FIGURE  4. 


PROBLEM  4.  Sketch  circles  in  several  positions 
as  suggested  in  Figure  4. 

PROBLEM  5.  Sketch  two  concentric  circles,  the 
diameters  of  which  are  3"  and  4£"  respectively. 

8.  The  Cylinder  in  Perspective.  Figure  2,  PLATE  VII, 
shows  the  circle  as  it  appears  on  three  faces  of  a  cube 
in  perspective.  The  axes  of  the  circular  planes,  carried 
through  to  the  corresponding  points  in  the  opposite 
faces  of  the  cube,  are  perpendicular  to  the  major  axes 
and  coincide  with  the  minor  axes  of  the  ellipses. 

It  is  sometimes  very  convenient  to  locate  a  cylinder 
in  perspective  by  first  drawing  a  square  prism,  for  then 
the  axis  of  the  cylinder  coincides  with  the  axis  of  the 
square  prism.  (See  Figure  4,  PLATE  XVI.)  The  direc- 
tion of  the  axis  of  the  cylinder  in  perspective  is  usually 
determined  first,  because  this  is  the  axis  of  revolution 


28  PERSPECTIVE  SKETCHING 

of  the  circular  planes  at  the  end  of  the  cylinder,  and 
from  this,  the  direction  of  the  major  axes  of  the  ellipses 
representing  these  circular  planes  is  determined  by  the 
rule  for  sketching  circles  in  perspective,  which  states 
that  the  major  axes  should  be  perpendicular  to  the  axes 
of  revolution. 

PROBLEM  6.  Make  a  sketch  showing  a  cylinder 
divided  into  circular  planes  as  in  PLATE  VIII. 

PROBLEM  7.  Sketch  cylinders,  2"  diam.,  4"  long,  in 
the  several  positions  suggested  in  PLATE  IX. 

9.  Estimating  Measurements.  In  order  to  sketch 
at  once  the  fundamental  lines  of  an  object,  it  is  neces- 
sary to  have  a  definite  idea  of  the  appearance  of  plane 
angles  from  different  points  of  view.  In  PLATE  II,  the 
different  planes  shown  in  the  drawing  of  the  square 
prism,  are  all  seen  at  varying  distances  above  and  below 
the  horizon  line.  The  retreating  edges  of  each  plane 
are  known  to  be  at  the  same  angle  to  the  picture-plane, 
yet  they  do  not  so  appear  in  the  drawing  because  the 
eye  is  seeing  each  plane  at  different  levels. 

The  foreshortening  of  lines  and  the  perspective 
value  of  angles  may  be  determined  by  drawing  a  circle, 
dividing  it  into  any  number  of  parts,  each  representing 
a  definite  number  of  degrees,  and  then  placing  it  in  a 
horizontal  position,  obliquely  to  the  line  of  vision,  at 
any  distance  below  the  eye.  The  upper  half  of  Figure 
5,  (which  is  an  orthographic  projection  of  such  a  circle) 
may  be  assumed  to  be  a  perspective  view  of  a  circle. 
The  diameter,  0-180,  is  parallel  with  the  picture-plane, 
and  the  vanishing-point  may  be  considered  as  being  at 
an  infinite  distance  from  the  observer.  The  long  axis 
represents  the  length  of  the  line  when  parallel  with  the 
picture  plane ;  the  short  axis  determines  the  extreme 
foreshortening  of  the  line  when  perpendicular  to  the 
picture-plane.  The  divisions  from  o  to  go  show  the 


FROM  WORKING  DRAWINGS 


FIGURE  5. 


varying  lengths,  or  foreshortening,  of  the  radii  of  the 
circle  as  they  appear  at  different  angles  to  the  picture- 
plane. 

The  upper  portion  of  Figure  5  may  be  called  an 
"ELLIPTICAL"  or  "PERSPECTIVE  PROTRACTOR"  which 
can  be  utilized  as  a  means  of  approximately  measuring 
plane  angles  and  foreshortening  of  lines  and  plane 
angles.  For  this  purpose,  it  is  of  great  assistance  in 
sketching  from  working  drawings,  to  have  such  an 
elliptical  protractor,  which  may  be  cut  from  cardboard, 
graduated  into  15  degrees  divisions.  The  proportion 
of  the  length  of  the  major  axis  to  that  of  the  minor  axis 
in  an  elliptical  protractor  giving  satisfactory  picture 


30 


PERSPECTIVE  SKETCHING 


FIGURE  6. 


results  in  perspective  is  about  one  to  three.  Figure  6, 
illustrates  the  projection  of  the  circular  protractor  in 
order  to  obtain  an  elliptical  protractor  having  these 
proportions. 

10.  The  Elliptical  Protractor.  FIGURE  6.  To  make 
an  elliptical  protractor  for  use  in  perspective  sketching, 
lay  out  the  major  and  minor  axes,  ab  and  cd,  and,  at 
the  intersection  of  these  axes  for  a  center,  draw  circles 
with  the  axes  as  diameters.  Divide  the  outer  circle 
into  24  equal  parts,  the  spaces  each  measuring  an  angle 
of  15  degrees.  Radial  lines  from  these  divisions  on  the 
outer  circle  divide  the  inner  circle  into  the  same  number 


FROM  WORKING  DRAWINGS 


31 


FIGURE  7. 

of  parts.  If  from  the  points  on  the  outer  circle,  vertical 
projection  lines  are  drawn,  intersecting  horizontal  pro- 
jection lines  from  corresponding  points  where  the  radial 
lines  divide  the  inner  circle,  the  intersections  will  give 
the  points  through  which  the  circumference  of  the 
ellipse  may  be  drawn.  The  intersections  of  these  hori- 
zontal and  vertical  projection  lines  are  the  division 
points  on  the  circumference  of  the  ellipse  for  each  angle 
of  15  degrees  which  should  be  numbered  from  o  to  180. 
By  dividing  the  major  axis  into  a  certain  number  of 
equal  units  and  then  dividing  each  radius  of  the  ellipse 
into  the  same  number  of  equal  units  as  in  the  horizontal 
radii,  the  approximate  foreshortening  of  each  radius 
and  any  fractional  part  of  that  radius  at  any  given  angle 
may  be  determined. 

Figures  7,  8  and  9  show  the  appearance  of  a  cube  in 
three  different  positions  at  different  angles  to  the  picture 
plane,  located  by  using  the  elliptical,  or  as  we  shall  call 
it  hereafter,  the  perspective  protractor.  Figure  7  shows 
the  cube  with  the  front  face  at  an  angle  of  15  degrees 


32 


PERSPECTIVE  SKETCHING 


FIGURE  8. 

to  the  picture-plane.  The  visible  face  perpendicular  to 
the  front  face  at  15  degrees  will  thus  be  at  75  degrees 
to  the  picture-plane  or  105  degrees  on  the  protractor. 
Figure^S,  shows  the  cube  with  the  front  face  turned 


FIGURE  9. 


FROM  WORKING  DRAWINGS 


33 


FIGURE  10. 

farther  away,  an  at  angle  of  30  degrees,  thus  making 
the  vertical  face  which  is  perpendicular  to  the  front  face 
at  an  angle  of  60  degrees  to  the  picture-plane. 
Figure  9,  shows  both  faces  of  the  cube  at  45  degrees 
to  the  picture-plane.  This  position  shows  the  greatest 
possible  perspective  width  of  a  cube,  and  is  the  one 
usually  assumed  for  the  perspective  position  of  a  cube 
used  as  a  unit  in  developing  a  perspective  sketch  from 
a  working  drawing.  If  it  is  required  to  make  a  drawing 
which  has  lines  longer  than  those  of  the  protractor,  a 
parallel  perspective  circle  may  be  sketched  which  shall 
be  proportionately  larger  than  the  protractor,  but  this 
is  not  altogether  necessary,  for  the  perspective  angles 
having  been  obtained  by  using  the  perspective  pro- 
tractor, the  line  may  be  extended  a  distance  proportion- 
ate to  the  length  of  the  lines  of  the  protractor. 
This  is  illustrated  in  Figure  10  which  represents  a  draw- 
ing of  a  rectangular  prism  which  is'as  high  as  it  is  wide 
and  whose  length  is  twice  that  of  its  height.  It  must  be 


34 


PERSPECTIVE  SKETCHING 


FIGURE  11. 

remembered  that  all  measurements  are  wholly  approxi- 
mate, and  that  the  protractor  is  used  simply  as  a  help 
in  estimating  the  perspective  foreshortening  and  posi- 
tion of  lines  and  angles.  In  all  perspective  sketching, 
good  judgment  in  estimating  measurements  is  the  im- 
portant thing  to  be  emphasized. 

11.  The  Rectangular  Prism  in  Perspective.  PROB- 
LEM 8.  Sketch  a  rectangular  prism  in  perspective 
similar  to  that  in  Figure  10,  in  two  different  positions. 
The  proportions  may  be  6"  by  2"  by  1". 

PROBLEM  9.  Make  a  sketch  in  perspective  from 
the  working  drawing  in  Figure  1 1 .  This  working  draw- 
ing represents  a  rectangular  prism  through  the  center 
of  which  is  a  rectangular  hole.  By  extending  the  lines 
representing  the  hole  to  the  outside  rectangle  in  the 
front  view,  there  are  two  series  of  parallel  lines,  one  a 
vertical  series,  and  the  other  a  horizontal  series.  If  the 
sketch  is  started  by  drawing  the  front  vertical  edge, 
(See  Figure  1,  PLATE  X)  the  location  of  each  of  the 
lines  in  the  horizontal  series  may  be  measured  on  this 


FROM  WORKING   DRAWINGS 


35 


PLATE  X. 


36 


PERSPECTIVE  SKETCHING 


/  * 

r-*?- 


'& 


rn 


-  fr-- 


fef 


iMoO 
__J|    ±    i 


FIGURE  12. 


edge  and  the  lines  sketched  retreating  to  a  common 
vanishing-point  which  is  at  an  indefinite  distance  away 
from  this  nearest  edge.  The  position  of  each  line  of  the 
vertical  series  may  be  measured  on  the  upper  edge  of 
the  front  face  and  these  lines  sketched  in  their  proper 
locations.  The  lines  of  the  back  face  may  be  deter- 
mined in  the  same  way.  The  sketch  may  be  completed 
by  erasing  the  construction  lines  and  strengthening 
the  outlines  of  the  prism  and  hole. 

PROBLEM  10.  Make  a  sketch  of  the  grooved  block 
from  the  working  drawing  of  the  block,  Figure  12.  This 
is  another  good  illustration  of  measuring  by  means  of 
parallel  lines,  for  the  lines  of  the  groove  are  parallel  with 
the  outside  edges  of  the  block.  After  sketching  the 
rectangular  block,  the  lines  of  the  groove  may  be  located 
on  the  end  of  the  block  and  the  groove  sketched  as 
shown  in  Figures  2  and  3,  PLATE  X. 

PLATE  XI.  Figure  1 ,  represents  a  rectangular  card 
resting  on  a  cube,  the  vertical  faces  of  which  are  at  45 
degrees  to  the  picture-plane.  One  end  of  the  card 
rests  on  the  same  plane  as  that  of  the  base  of  the  cube. 
After  the  cube  is  sketched,  the  location  of  points  /  and 
3  of  the  card  may  be  determined  by  extending  the  lines, 
ab  and  a'b' ,  in  the  direction  in  which  it  is  desired  to 


FROM  WORKING  DRAWINGS 


37 


PLATE  XI. 


38 


PERSPECTIVE  SKETCHING 


FIGURE  13. 

place  the  lower  edge  of  the  card.  When  point  /  is 
located,  then  sketch  the  line,  i-j  converging  with  aa' 
and  66'.  Sketch  the  side  lines  of  the  card  from  i  through 
c  and  from  j  through  c'  and  locate  2  and  4  by  cutting 
the  lines,  102  and  y'  4  with  a  line  converging  with  cc', 
which  completes  the  drawing  of  the  card. 

PROBLEM  11.  Sketch  a  group  consisting  of  a  cube 
and  two  rectangular  prisms  as  suggested  in  Figure  2, 
PLATE  XI.  The  sketch  of  the  square  card  in  this  figure 
may  be  considered  the  base  of  the  cube. 

Figure  13  is  a  good  illustration  showing  the  use 
of  a  rectangular  prism  as  a  unit  in  making  a  perspec- 


FROM  WORKING  DRAWINGS 


39 


t! 


FIGURE  14. 

tive  sketch  of  the  table.  After  sketching  the  prism, 
the  positions  of  the  bases  of  the  legs  of  the  table  may  be 
located  by  measuring  the  lines  ab  and  ac,  and  sketching 
each  system  of  retreating  parallel  lines. 

PROBLEM  12.     Sketch  in  perspective,   the  bench 
represented  by  the  working  drawing  in  Figure  14. 


12.  The  Equilateral  Triangle  in  Perspective.  PLATE 
XII.  Figures  1  and  2  are  the  working  drawings  of  an 
equilateral  triangular  prism  to  be  sketched  in  perspective 
with  the  end  faces  at  an  angle  of  45  degrees  to  the 
picture-plane.  After  locating  the  line,  ac,  in  Figure  3, 
with  the  perspective  protractor,  draw  ex,  equal  in 
height  to  the  altitude  of  the  triangle.  The  point,  e,  is 
at  the  perspective  center  of  ac,  and  a  perpendicular 
from  this  point,  parallel  with  ex,  and  cut  by  the  line, 
xy,  converging  with  ac,  determines  the  altitude  of  the 
triangle,  eb.  Connecting  ab  and  ac  completes  the 
equilateral  triangle  in  perspective.  The  line,  cc',  is 
perpendicular  to  ca,  and  the  proportions  of  the  sketch 
show  it  to  be  about  twice  as  long.  Retreating  converg- 
ing lines  from  a,  b,  and  c  of  the  front  end  of  the  block 
aid  in  determining  the  position  of  the  points,  a' ,  b' ,  and 
c'  of  the  farther  end  of  the  block.  Figure  5  shows  the 
end  of  a  triangular  prism  of  which  the  perspective  is 


40 


PERSPECTIVE  SKETCHING 


PLATE  XII. 


FROM  WORKING  DRAWINGS  41 

shown  parallel  with  the  ground-plane  in  Figure  6.  Using 
the  perspective  protractor,  the  positions  of  the  corners, 
a,  b  and  c,  may  be  determined,  and  the  lines,  ab,  be  and 
ca  represent  the  perspective  view  of  the  lower  end  of  the 
block.  The  block  may  be  completed  as  shown  in 
Figure  6,  by  drawing  vertical  lines  from  a,  b  and  c,  and 
the  lines,  a'b' ',  b'c'  and  c'a',  of  the  top  of  the  block. 

PROBLEM  13.  Sketch  in  perspective,  two  equi- 
lateral triangular  prisms  of  equal  size,  following  the  con- 
struction and  in  the  positions  shown  in  the  sketch  in  the 
upper  part  of  PLATE  XII. 

13.  The  Hexagon  in  Perspective.  PLATE  XIII.  Fig- 
ure 1  of  this  plate  illustrates  some  of  the  peculiar  proper- 
ties of  the  hexagon.  There  are  six  equilateral  triangles 
in  this  figure.  The  diagonals  are  twice  the  length  of  a 
side,  and  each  diagonal  is  parallel  with  two  opposite 
sides.  There  are  three  series  of  parallel  lines,  and  the 
lines  connecting  the  ends  of  opposite  sides  divide  the 
diagonal  equally  between  the  center  and  the  ends  of  the 
diagonal  to  which  they  are  perpendicular.  It  may  be 
inscribed  in  a  circle,  which  makes  the  perspective 
sketching  of  the  hexagon  a  simple  matter. 

Figure  2  is  a  perspective  sketch  of  the  hexagon 
directly  in  front  of  the  observer  and  located  below  the 
level  of  the  eye.  The  diagonal,  eb,  is  parallel  with  the 
picture-plane.  Each  series  of  retreating  parallel  lines, 
ab,  fc,  ed,  and  fe,  ad,  be,  should  show  slight  convergence 
toward  the  vanishing-points  in  the  direction  taken 
by  the  series. 

Figure  3  illustrates  the  perspective  of  a  hexagona 
prism  resting  on  an  end  face  with  the  front  side  parallel 
with  the  picture-plane. 

In  Figures  4  and  5,  the  hexagon  is  built  up  from 
one  of  its  sides,  af.  After  locating  af  with  the  perspec- 
tive protractor,  sketch  the  height  lines,  ac  and  fd,  of 


42 


PERSPECTIVE  SKETCHING 


PLATE  XIII. 


FROM  WORKING  DRAWINGS 


43 


B 

N 

J 


FIGURE  15. 

the  rectangular  figure,  acdf,  inscribed  in  the  hexagon 
in  Figure  1,  and  finish  the  perspective  of  the  rectangle 
by  sketching  cd  converging  with  af.  Sketch  the  diag- 
onals, ad  and  cf,  obtaining  the  perspective  center,  o, 
of  the  rectangle  through  which  the  line,  hg,  is  sketched, 
converging  with  af  and  cd.  In  Figure  1,  the  spaces, 
eg>  g°>  °h,  and  hb,  are  equal,  but  in  sketching  the  hexa- 
gon, the  spaces  should  each  have  slight  foreshortening 
as  they  retreat  from  the  picture-plane.  This  fore- 
shortening may  be  taken  care  of  in  Figure  5  by  making 
bh  a  little  longer  than  ho  and  ge  a  little  shorter  than  og. 
The  lines  connecting  ab,  be,  de,  and  ef  finish  the  sketch 
of  the  hexagon. 

In  Figure  6,  the  axis  of  the  hexagon  is  determined 
and  a  perspective  circle  sketched  which  will  circum- 
scribe the  hexagon.  The  major  axis  of  the  perspective 
circle  is  one  of  the  diagonals,  and  locates  two  corners, 
a  and  d,  of  the  hexagon.  The  corners,  /  and  c,  are  lo- 
cated by  sketching  parallel  vertical  lines  from  a  and  d. 
Sketch  the  diagonals,  ad  and  cf,  and  locate  e  and  b  by 


44 


PERSPECTIVE  SKETCHING 


F/g.l- 


PLATE  XIV. 


FROM  WORKING  DRAWINGS  45 

drawing  a  line  through  o  which  converges  with  af  and 
cd. 

PROBLEM  14.  Sketch  a  hexagonal  prism  in  per- 
spective, following  construction  represented  in  Figure 
6,  PLATE  XIII. 

PROBLEM  15.  Sketch  the  hexagonal  nut  repre- 
sented in  Figure  15,  in  three  different  positions. 

14.  The  Square  Pyramid  in  Perspective.  PLATE  XIV. 
After  sketching  the  base  of  the  pyramid  in  any  desired 
position,  the  location  of  the  axis  may  be  approximately 
determined  by  sketching  it  perpendicular  to  the  longer 
diagonal  of  the  base  at  the  intersection  of  the  diagonals. 
The  foreshortening  of  the  axis  may  be  measured  by 
estimating  its  length  in  comparison  with  the  side  of  a 
rectangular  plane  passing  through  the  axis,  perpendicu- 
lar to  the  base,  and  the  size  of  this  plane  determined 
by  its  proportion  to  the  base.  Figure  3  represents  the 
pyramid  with  the  edges,  ab  and  be,  of  the  base  at  45 
degrees  to  the  picture-plane.  Sketching  the  base,  abed, 
and  its  diagonals,  the  axis,  oe,  is  found  to  be  perpendicu- 
lar to  ac  at  e.  Figure  4  represents  the  pyramid  resting 
on  one  of  the  triangular  faces,  aob. 

PROBLEM  16.  Make  a  sketch  of  a  square  pyramid 
writh  the  base  resting  on  the  edge  of  a  rectangular  prism 
as  in  Figure  5,  PLATE  XTV. 


46 


PERSPECTIVE  SKETCHING 


TJje  Cor/6  In 


•     — 
.  3- 


PLATE  XV. 


FROM  WORKING  DRAWINGS  47 

15.  The  Cone  in  Perspective.  PLATE  XV.  In  sketch- 
ing the  cone  in  perspective,  the  same  construction  that 
is  used  in  sketching  the  cylinder  is  followed.  After 
determining  the  direction  of  the  axis  of  the  cylinder, 
and  sketching  the  ellipse  representing  the  base,  the  side 
lines  are  sketched  from  the  apex  at  the  end  of  the  axis 
and  always  tangent  to  the  circumference  at  the  base. 

PROBLEM  17.  Sketch  a  cone  with  a  base  2£" 
diameter,  height  4$'',  in  the  several  positions  suggested 
in  PLATE  XV. 


48  PERSPECTIVE  SKETCHING 

/ 

16.  Sketching  from  Working  Drawings.  It  must  be 
remembered  that  the  working  drawing  represents  a 
collection  of  facts  shown  in  several  views,  each  relating 
to  a  certain  number  of  these  facts.  The  drawing  is  made 
according  to  the  principles  of  orthographic  projection. 
The  views  are  really  projections,  not  pictures,  and  as 
many  such  projections  must  be  made  as  are  necessary 
to  represent  the  object  so  clearly  and  completely  that 
a  workman  may  make  it  exactly  as  it  is  intended  to  be 
without  any  other  directions  than  those  contained  in 
the  working  drawings.  Each  projection  shows  its  par- 
ticular side  of  the  object.  While  several  projections 
are  necessary  to  completely  represent  the  object  in  a 
working  drawing,  the  perspective  sketch  must  have  all 
the  facts  contained  in  these  different  projections 
brought  together  and  represented  in  one  view.  This 
is  illustrated  in  all  of  the  previous  problems.  A  knowl- 
edge of  typical  forms,  their  appearance  in  working 
drawings,  and  also  their  appearance  in  perspective  is 
necessary,  for  it  is  the  only  available  requisite  infor- 
mation of  value  in  sketching.  A  working  drawing 
representing  any  machine  or  architectural  construction, 
can  be  analyzed  into  units  of  the  familiar  typical  forms, 
as  the  square  prism,  cube,  rectangular  prism,  or  cylin- 
der. Usually,  there  are  but  three  directions  taken  by 
the  different  series  of  parallel  lines  in  the  unit,  and 
generally  one  of  these  series  consists  of  vertical  lines, 
so  that  the  other  two  series  are  the  ones  to  be  consid- 
ered for  convergence  and  perspective  foreshortening 
of  lines.  The  perspective  sketch  may  be  located  by 
drawing  the  outside  lines  of  the  series,  and  if  the  con- 
vergence and  foreshortening  are  taken  care  of  in  these 
outside  lines,  the  position  of  all  other  lines  parallel  with 
and  lying  between  them,  is  more  easily  determined. 
It  is  therefore  necessary,  when  sketching  from  working 
drawings  made  up  of  several  forms,  to  have  a  method 
which  will  lead  quickly  to  accurate  results. 


FROM  WORKING  DRAWINGS  49 

The  problems  already  considered  are  simple  work- 
ing drawings  analyzed  into  a  single  unit,  such  as  the 
cube  or  rectangular  prism.  This  unit  has  been  used  as 
a  basis  for  the  foreshortening  of  plane  surfaces  and  lines. 
By  cutting  the  surfaces  with  several  series  of  parallel 
lines  and  taking  away  from  the  unit  certain  portions, 
the  changes  necessary  to  give  the  final  sketch  result 
have  been  obtained.  When  a  working  drawing  can  be 
analyzed  into  a  number  of  different  typical  forms,  then 
one  of  these  forms  should  be  used  as  a  unit  in  deter- 
mining the  proportions  of  the  sketch.  This  division  of 
the  working  drawing  requires  careful  judgment  in  re- 
ducing is  to  the  simplest  typical  forms  in  which  the 
perspective  principles  can  be  readily  carried  out. 


50 


PERSPECTIVE  SKETCHING 


<  
f)"                 » 

<o     , 

)  "  J 

^ 

—  A 

2. 

1 

/ 

\ 

* 

V 

i 

«N 

c 

fy 

fe 

p* 

r 

1 

FIGURE  16. 


PROBLEM  18. 

Hollow  Sleeve.  PLATE  XVI.  The  working  draw- 
ings, Figures  1  and  2,  represent  a  hollow  sleeve  which 
may  be  analyzed  into  the  forms  of  a  square  prism  and 
cylinder.  The  lines,  1-2  and  3-4,  in  Figure  2,  form  a 
rectangle  which  is  one  side  of  a  square  prism,  the  end  of 
which  is  shown  in  Figure  1 .  Figure  3  is  the  perspective 
sketch  of  this  square  prism  with  the  vertical  faces  at  45 
degrees  to  the  picture-plane.  The  axis,  ab,  of  the  cylinder 
is  located  by  connecting  the  intersection  of  the  diagonals 
in  the  top  and  lower  faces.  The  square  prism  at  the 
center  of  the  sleeve  is  determined  by  measuring  its 
location  on  the  front  vertical  edge  and  sketching  a 
series  of  parallel  lines  defining  its  position.  The  major 
axes,  cd,  of  the  ellipses  representing  the  various  cir- 
cular planes  of  the  cylinder,  coincide  with  the  long 
diagonals  of  the  square  planes  in  this  figure.  Figure 
4  is  a  sketch  of  the  sleeve  in  a  horizontal  position. 
Sketch  the  square  prism  enclosing  the  sleeve  and 
locate  the  axis  of  the  prism  which  coincides  with  the  axis 
of  the  sleeve.  Note  that  the  major  axes,  cd,  of  the  ellipses 
are  all  perpendicular  to  the  axis,  ab,  and  also  that 
these  major  axes  do  not  coincide  with  either  diameters 
or  diagonals  of  the  square. 

PROBLEM  19.  Sketch  the  Hollow  Sleeve,  Figure 
16,  showing  the  axis  at  30  degrees  to  the  picture-plane 
and  parallel  with  the  ground-plane. 


FROM  WORKING  DRAWINGS 


51 


PLATE  XVI. 


PERSPECTIVE  SKETCHING 


a. 


3 


.1 


</' 


FIGURE  17. 


PROBLEM  20. 


Bush  for  Bearings.  PLATE  XVII.  Sketch  the  con- 
struction lines  represented  in  the  working  drawing, 
Figure  17,  and  note  the  series  of  parallel  lines  thus  in- 
dicated. In  Figure  1,  the  rectangular  prism  which 
serves  as  a  unit  for  this  sketch,  is  located  by  the  per- 
spective protractor  at  45  degrees  to  the  picture-plane. 
The  construction  lines  in  the  working  drawing  divide 
the  front  view  into  a  number  of  squares  which  are  of 
assistance  in  making  the  perspective  sketch.  Continue 
the  vertical  lines,  2-7  and  4-5,  and  obtain  the  vertical 
square  in  perspective  by  sketching  the  line,  8-7  converg- 
ing with  ab.  From  o,  the  axis,  oo' ' ,  of  the  cylinder 
converges  with  the  lines,  aa'  and  bbf,  of  the  prism. 
After  locating  the  major  axis  of  the  ellipse,  perpen- 
dicular to  the  axis  of  the  cylinder,  sketch  the  ellipse 
representing  the  end  of  the  circular  groove  in  the  bear- 
ing. Converging  lines  from  i  and  5  locate  points,  /' 


FROM  WORKING  DRAWINGS 


53 


-Bush  for  Bearings 


PLATE  XVII. 


54 


PERSPECTIVE  SKETCHING 


FIGURE  18. 

and  5',  on  the  farther  edge  of  the  prism.  The  sketch 
is  completed  by  drawing  any  other  necessary  lines, 
care  being  taken  to  see  that  they  converge  with  the 
outside  lines  with  which  they  are  parallel  in  the  object. 

PROBLEM  21. 

Make  a  sketch  of  Bearing,  Figure  18,  with  the 
longer  side  of  the  base  at  45  degrees  to  the  picture- 
plane. 

PROBLEM  22. 

Grooved  Block.  PLATE  XVIII.  This  plate  is  an 
excellent  example  of  measuring  by  the  intersections  of 
two  series  of  parallel  lines.  The  positions  of  the  corners 
of  the  groove  with  inclined  sides  are  readily  measured 
on  the  nearest  line,  (ab,  Figure  3).  The  converging 
lines  from  the  points  on  this  line,  intersected  by  the 
construction  line  parallel  with  ab,  determine  the  location 
of  the  corners  of  the  groove  in  the  end  face  of  the  block. 
The  rectangular  grooves  are  located  in  the  same  way 
in  the  top  of  the  block. 

PROBLEM  23. 

Sketch  the  grooved  block  with  the  upper  face 
showing  the  inclined  groove  and  parallel  with  the 
ground-plane. 


FROM  WORKING  DRAWINGS 


55 


-  rr-n-'r  iTi-"-"n  •  rit  -i''  "T  -nff  r" ili 


Grooved 


<v 


\ 

,--- 

""" 

—  -~i—  I 

H- 

1 

1 
/ 

* 

1 

V 

1 

I 

i  

L. 

-£-* 

1 
u:f- 

PLATE  XVIII. 


56 


PERSPECTIVE  SKETCHING 


FIGURE  19. 


PROBLEM  24. 

V  Block.  PLATE  XIX.  Sketch  the  construction 
lines  shown  in  Figure  19,  and  measure  distances  on  the 
nearest  line  of  the  rectangular  prism  in  Figure  1  of  this 
plate.  Note  the  direction  of  the  axes  of  the  circular 
grooves  in  fixing  the  direction  of  the  major  axes  of 
the  ellipses  representing  the  ends  of  the  grooves. 

PROBLEM  25.  Sketch  the  V  Block,  Figure  19, 
in  the  two  positions  suggested  in  Figure  2  of  this  plate. 


FROM  WORKING  DRAWINGS 


57 


PLATE  XIX. 


58  PERSPECTIVE  SKETCHING 

PROBLEM  26. 

The  Crank.  PLATE  XX.  When  the  cylinder  is  the 
most  common  form  in  a  working  drawing,  it  is  some- 
times more  convenient  to  first  determine  the  position  of 
the  axes  of  the  cylinders  and  build  up  the  sketch  from 
the  cylindrical  forms.  The  analysis  of  Figures  1  and  2 
shows  two  cylinders  connected  by  a  rectangular  prism, 
of  which  the  lines,  ae,  ef,  fb,  and  ba,  form  one  side  in 
Figure  2.  The  lines,  ae,  and  bf,  in  Figure  2,  are  tangent 
to  the  two  cylinders  at  the  ends  of  their  horizontal 
diameters.  The  points,  I,  2,  j,  and  4,  in  Figure  \ ,  are 
corners  of  a  square  enclosing  the  circle,  which  in  Figure 
3,  determines  the  size  of  the  ellipse  sketched  tangent 
to  the  sides  of  the  perspective  square.  The  end  faces 
of  the  cylinders  are  parallel  with  the  ground-plane,  and 
so  the  major  axes  of  the  ellipses  representing  them  will 
appear  parallel  with  the  picture-plane.  The  axis,  abc, 
is  perpendicular  to  the  major  axis  of  the  ellipse  at  the 
base  of  the  cylinder.  The  axis,  def,  is  parallel  with  abc, 
and  converging  lines  from  a  and  b  locate  the  centers, 
e  and  /.  Sketch  the  ellipses  and  the  tangents  connect- 
ing them.  The  small  ellipses  at  a  and  d  represent  holes 
in  the  crank,  and  are  concentric  with  the  larger  ellipses 
enclosing  them. 


FROM  WORKING  DRAWINGS 


59 


PLATE  XX. 


6O 


PERSPECTIVE  SKETCHING 


FIGURE  20. 


PROBLEM  27. 


Sketch  the  Shaft  Support,  Figure  20,  with  the 
cylindrical  part  turned  at  45  degrees  to  the  picture-plane 
and  lying  in  a  horizontal  position. 

PROBLEM  28. 

Offset  Standard.  PLATE  XXI.  Sketch  the  end,  abed, 
of  the  prism  and  the  series  of  parallel  lines  indicated 
by  the  construction  lines  in  Figure  2.  The  axis,  oo', 
of  the  cylinder  is  parallel  with  the  line,  hij,  of  the  base. 
Lines  converging  with  the  axis,  oo',  and  tangent  to  the 
ellipse  on  the  end  of  the  prism,  determine  the  size  of 
the  ellipses  at  i,  2,  and  j.  Curved  lines  from  i,  k,  and  / 
sketched  tangent  to  the  ellipses,  connect  the  cylinder 
with  the  base. 


FROM  WORKING  DRAWINGS 


61 


-Offset 


Hg./ 


PLATE  XXI. 


PERSPECTIVE  SKETCHING 


FIGURE  21. 

PROBLEM  29.  Sketch  the  Hinge-Block,  Figure  21, 
with  the  face  containing  the  T  shaped  groove  turned 
at  30  degrees  to  the  picture-plane  and  perpendicular 
to  the  ground-plane. 

PROBLEM  30. 

Angle  Brace.  PLATE  XXII.  Enclose  the  working 
drawings,  Figures  1  and  2,  in  construction  lines  and  note 
each  series  of  parallel  lines.  Sketch  the  perspective 
outline  of  the  prism,  locate  the  axes  of  the  cylindrical 
forms  at  each  end  of  the  brace,  and  draw  the  ellipses 
representing  the  circular  ends.  The  diagonal  indicated 
is  a  part  of  the  major  axes  of  the  ellipses  representing 
the  curved  part  connecting  the  ends  of  the  brace. 

PROBLEM  3 1 .  Sketch  the  Angle  Brace  showing  the 
curved  part  parallel  with  the  ground-plane. 


FROM  WORKING  DRAWINGS 


63 


-Anqledrace- 


1  «       > 

s&- 

*  «  ^ 

1* 

^ 

7"' 
tL-4 

T 

//      ( 

^1 

i 

>    i 

PLATE  XXII. 


64 


PERSPECTIVE  SKETCHING 


FIGURE  22. 
i 

PROBLEM  32.  Sketch  the  Connecting  Rod  End, 
Figure  22,  with  the  axis  of  the  cylindrical  part  at  an 
angle  of  45  degrees  to  the  picture-plane  and  parallel 
with  the  ground-plane. 

PROBLEM  33. 

Intersecting  Square  Prisms.  PLATE  XXIII.  The 
construction  shown  in  this  plate  is  another  good  ex- 
ample of  measuring  by  the  intersections  of  the  two  series 
of  parallel  lines.  After  sketching  the  outline  of  the 
square  prism  enclosing  the  working  drawing,  the  ends 
of  the  prisms,  i,  2,  j,  4,  and  13,  14,  15,  16,  should  be 
located.  From  these  ends,  each  prism  may  be  com- 
pleted, and  the  lines  of  intersection  located  by 
connecting  the  points  at  the  intersection  of  the  edges  of 
the  prisms. 


FROM  WORKING  DRAWINGS 


65 


PLATE  XXIII. 


66 


PERSPECTIVE  SKETCHING 


FIGURE  23. 


FIGURE  24. 


FROM  WORKING  DRAWINGS 


67 


FIGURE  25. 

PROBLEM  34.  Sketch  the  Intersecting  Pyramid 
and  Prism,  Figure  24,  with  the  axis  of  the  pyramid  at 
30  de'grees  to  the  picture-plane  and  parallel  with  the 
ground-plane. 

PROBLEM  35.  Sketch  the  Intersecting  Cylinders, 
Figure  25,  with  the  axes  of  the  horizontal  cylinders 
parallel  with  the  ground-plane  and  turned  at  30  de- 
grees to  the  picture-plane. 

PROBLEM  36.  Sketch  the  Bolt,  Figure  23,  with 
the  axis  at  30  degrees  to  the  picture-plane  and  parallel 
with  the  ground-plane. 


68 


PERSPECTIVE  SKETCHING 


FIGURE  26 


PROBLEM  37. 


Crank.  PLATE  XXIV.  Sketch  the  axis,  oo",  of  the 
larger  cylinder  at  60  degrees,  and  the  line,  1-2,  at  30 
degrees  to  the  picture-plane,  measuring  with  the  per- 
spective protractor.  The  perspective  circle,  abed,  de- 
termines the  foreshortened  appearance  of  the  diameter, 
1-2.  The  major  axis,  ef,  of  the  ellipse  is  perpendicular 
to  the  axis  of  the  cylinder  at  o,  and  the  ellipse  passes 
through  i  and  2  of  the  measuring  ellipse,  acbd.  The 
axis,  gg",  of  the  smaller  cylinder  converges  with  the 
axis,  oo",  of  the  larger  cylinder.  Parallel  lines  from  o' 
and  o"  determine  the  location  of  g'  and  g",  and  the 
major  axes  of  the  ellipses  at  the  small  end  of  the  crank 
are  perpendicular  to  the  axis  of  the  cylinder  at  points, 
g,  gf,  and  g".  Sketch  the  ellipses  and  the  tangents 
connecting  them.  Complete  the  drawing  by  sketching 
the  small  ellipses  at  o  and  g,  representing  the  holes 
in  the  crank,  concentric  with  the  larger  ellipses  enclos- 
ing them. 

PROBLEM  38.  Sketch  the  crank  with  the  axes  of 
the  cylinders  perpendicular  to .  the  ground-plane  and 
the  lines,  o'-gr,  at  45  degrees  to  the  picture-plane. 

PROBLEM  39.  Make  a  sketch  of  the  handle, 
Figure  26,  with  the  axis  of  the  hub  perpendicular  to  the 
ground-plane  and  the  axis  of  the  handle  at  30  degrees 
to  the  picture -plane. 


FROM  WORKING  DRAWINGS 


69 


PLATE  XXIV. 


7O 


PERSPECTIVE  SKETCHING 


FIGURE  27. 


PROBLEM  40. 

Pipe-Tee.  PLATE  XXV.  Sketch  the  outline  of  a 
rectangular  prism  that  will  enclose  the  Pipe-Tee  in 
Figures  1  and  2,  and  determine  the  location  and  direc- 
tion of  the  axes  of  the  cylinders  as  indicated  in  Figure  3 . 
From  these  axes  of  the  cylinders,  the  direction  of  the 
major  axes  of  the  various  ellipses  may  be  determined 
and  their  positions  located.  Finish  drawing,  following 
construction  suggested  in  Figure  3. 

PROBLEM  41.  Make  a  sketch  of  the  Pipe-Elbow, 
Figure  27,  resting  on  one  of  the  circular  ends  and  with 
the  other  end  turned  at  30  degrees  to  the  picture-plane. 


FROM  WORKING  DRAWINGS 


71 


PLATE  XXV. 


72 


PERSPECTIVE  SKETCHING 


FIGURE  28. 


PROBLEM  42. 

Shaft  Hanger.  PLATE  XXVI.  This  object  should 
be  sketched  in  a  position  above  the  eye-level,  as 
represented  in  Figure  3.  Sketch  the  construction  lines 
and  note  the  directions  of  the  axes  of  the  cylindrical 
forms  contained  in  the  working  drawing. 

PROBLEM  43.  Sketch  in  a  position  above  the  eye- 
level,  the  Pipe  Hanger  represented  by  the  working 
drawing  in  Figure  28. 


FROM  WORKING  DRAWINGS 


73 


PLATE  XXVI. 


74 


PERSPECTIVE  SKETCHING 


PLATE  XXVII. 


FROM  WORKING  DRAWINGS 


75 


L//?/ yersa/  Coc/pting 

<5co/e  tfa/fs/ze 


FIGURE  29. 


PROBLEM  44. 


Tool  Post  and  Carriage.  PLATES  XXVII  and 
XXVIII.  This  is  more  complicated  than  the  previous 
problems,  and  the  working  drawing  should  be  care- 
fully analyzed  into  the  different  typical  forms.  Treat 
each  form  separately  in  the  sketch,  but  locate  it  in  its 


76 


PERSPECTIVE  SKETCHING 


*^^ 


PLATE  XXVIII. 


FROM  WORKING  DRAWINGS  77 

relative  position  with  the  other  forms,  carefully  con- 
sidering the  convergence  of  lines  and  the  proportion 
of  parts. 

PROBLEM  45.  Sketch  the  Universal  Coupling, 
Figure  29,  showing  the  axes  of  the  shafts  at  30  degrees 
to  the  picture-plane  and  parallel  with  the  ground- 
plane. 


A     000  039  771     1 


